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advanced_tools:group_theory:representation_theory:adjoint_representation

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advanced_tools:group_theory:representation_theory:adjoint_representation [2020/11/29 16:59]
edi [Concrete]
advanced_tools:group_theory:representation_theory:adjoint_representation [2020/12/26 22:49]
edi [Concrete]
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 **Example** **Example**
  
-The diagram below shows the defining representation of $SU(2)$ in its upper branch. To construct the adjoint representation,​ we use the Lie algebra as the representation space, as shown in the lower branch (red arrows). The group elements act on this space like $L'​=ULU^{-1}$ and the Lie-algebra elements like $L'​=[J,​L]$. It is possible to rewrite ​this representation such that it acts on 3-dimensional vectors (as opposed to 3x3 matrices) by regular matrix-vector multiplication.+The diagram below shows the defining representation of $SU(2)$ in its upper branch. To construct the adjoint representation,​ we use the Lie algebra as the representation space, as shown in the lower branch (red arrows). The group elements act on this space like $L'​=ULU^{-1}$ and the Lie-algebra elements ​act like $L'​=[J,​L]$. It is possible to rewrite ​the adjoint ​representation such that it acts on 3-dimensional vectors (as opposed to 2x2 matrices) by regular matrix-vector multiplication.
  
 [{{ :​advanced_tools:​group_theory:​representation_theory:​su2_adjoint.jpg?​nolink }}] [{{ :​advanced_tools:​group_theory:​representation_theory:​su2_adjoint.jpg?​nolink }}]
advanced_tools/group_theory/representation_theory/adjoint_representation.txt ยท Last modified: 2020/12/26 22:49 by edi